# Single Component Metropolis-Hastings (i.e. component-wise updating)

So, let's say I have the following 2-dimensional target distribution that I would like to sample from (a mixture of bivariate normal distributions) -

import numba
import numpy as np
import scipy.stats as stats
import seaborn as sns
import pandas as pd
import matplotlib.mlab as mlab
import matplotlib.pyplot as plt
%matplotlib inline

def targ_dist(x):

target = (stats.multivariate_normal.pdf(x,[0,0],[[1,0],
[0,1]])+stats.multivariate_normal.pdf(x,[-6,-6],[[1,0.9],
[0.9,1]])+stats.multivariate_normal.pdf(x,[4,4],[[1,-0.9],[-0.9,1]]))/3
return target


and the following proposal distribution (a bivariate random walk) -

def T(x,y,sigma):

return stats.multivariate_normal.pdf(y,x,[[sigma**2,0],[0,sigma**2]])


The following is the Metropolis Hastings code for updating the "entire" state in every iteration -

#Initialising

n_iter = 30000

# tuning parameter i.e. variance of proposal distribution
sigma = 2

# initial state
X = stats.uniform.rvs(loc=-5, scale=10, size=2, random_state=None)

# count number of acceptances
accept = 0

# store the samples
MHsamples = np.zeros((n_iter,2))

# MH sampler
for t in range(n_iter):

# proposals
Y = X+stats.norm.rvs(0,sigma,2)

# accept or reject
u = stats.uniform.rvs(loc=0, scale=1, size=1)

# acceptance probability
r = (targ_dist(Y)*T(Y,X,sigma))/(targ_dist(X)*T(X,Y,sigma))
if u < r:
X = Y
accept += 1
MHsamples[t] = X


However, I would like to update "per component" (i.e. component-wise updating) in every iteration. Is there a simple way of doing this?

Thank you for your help!

## 1 Answer

[I did not read your code. I'm answering the statistical question.]

Sure, you can do a Metropolis update on each of the generations from full conditional distributions; this would just be the extreme version of a Metropolis-within-Gibbs -- it's like Gibbs sampling because every one of the conditionals you want to sample from would be conditional on all the others (and it's an extreme form of M-within-G in the sense that every one of those full conditional draws is obtained via a Metropolis-Hastings step).

[You can also do numerous other variations on just sampling full conditionals -- you just have to establish the usual properties of the Markov Chain you need; this is usually reasonably straightforward to argue, but in some situations you can get yourself into a situation where it's possible to screw things up (where the properties you require won't necessarily hold) so you can't just assume something works.]

You should, however, consider the sampling scheme: it's better to have a reversible scheme than just to cycle through $\theta_1,\theta_2,...,\theta_k$ (for example you can randomize the order or you can go forward and then backward within each iteration -- those are examples of reversible schemes).

It's not necessarily ideal to work this way, but there's nothing stopping you doing it in general.

• Maybe this is well-known, but would you have a reference for the penultimate paragraph? – hejseb Oct 27 '17 at 20:08
• Unless I misunderstand something in the question here, component-wise updates means you're using a different proposal for each component, in which case you don't necessarily satisfy detailed balance unless it's explicitly constructed to be reversible. See Juho's answer here for explicit discussion of the issue with component-wise updating. This is usually thought of as a Gibbs-sampling issue (so mentions of the issue often occurs in discussion of it) but it would be a potential problem with component-wise updating in MH more generally. ...ctd – Glen_b Oct 27 '17 at 23:05
• ctd... Google on keywords like reversibility and detailed balance to find more information, but I think Juho's answer there is an excellent starting place (I hadn't seen it before just now, I came across it while looking for a good reference to offer you). XI'an's comment underneath it is a good short summary (the individual moves are reversible, the generally the composition is not). For additional discussion, see Wikipedia here, for example – Glen_b Oct 27 '17 at 23:05
• Here's a journal article type reference if that's of any help: Richard A. Levine (2005) A note on Markov chain Monte Carlo sweep strategies, Journal of Statistical Computation and Simulation, 75:4, 253-262, DOI: 10.1080/0094965042000223671 ... – Glen_b Oct 28 '17 at 5:07